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Results tagged with combinatorial-designs
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user 27829
Design theory is the subfield of combinatorics concerning the existence and construction of highly symmetric arrangements. Finite projective planes, latin squares, and Steiner triple systems are examples of designs.
4
votes
Block design question
Sounds like similar to a forbidden configuration problem in extremal set theory, hypergraph theory, and design theory. I don't know if exactly the same problem has been studied in one of those fields …
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3
votes
Accepted
Resolvable designs from projective space
The fact that every odd dimensional projective geometry ${\rm PG}(n,2)$ over $\mathbb{F}_2$ admits a line parallelism (i.e., the Steiner $2$-design formed by the points and lines of ${\rm PG}(n,2)$ wi …
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8
votes
What is the largest number of k-element subsets of a given n-element set S such that…
You're asking what the number of blocks of a maximum packing is.
An ordered pair $(S, \mathcal{B})$ of a finite set $S$ of cardinality $\vert S \vert = v$ and a finite set $\mathcal{B}$ of $k$-subset …
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13
votes
Accepted
On the Steiner system $S(4,5,11)$
Unfortunately, no. It is known that the maximum number of mutually disjoint $S(4,5,11)$s on the same point set is $2$. Any such pair are always isomorphic. So, you can't find $7$ disjoint copies of an …
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7
votes
Accepted
"Codes" in which a group of words are pairwise different at a certain position
It is called perfect hash families in the design theory and computer science literature.
A perfect hash family PHF$(N; k, v, t)$ is an $N \times k$ array on $v$ symbols with $v \geq t$,
where for eve …
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8
votes
Accepted
covering designs of the form $(v,k,2)$
Edit: The possible "gap" of sort in Caro and Yuster's proof of their upper bound has just been fixed! See Ben Barber's comment below (and his joint paper with Daniela Kühn, Allan Lo and Deryk Osthus o …
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3
votes
Known results on cyclic difference sets
I'm not sure exactly what you mean because if you prove that there exists a cylcic $(v, k, \lambda)$-difference set for all $v$ except those that are excluded by known nonexistence results, you actual …
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6
votes
Accepted
Is there an infinite number of combinatorial designs with $r=\lambda^{2}$
So I read your edit, and here's the answer: Yes. Infinitely many of them exist.
Anyway, if you only need a $2$-design with $r = \lambda^2$ which has at least one pair of blocks intersecting each othe …
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2
votes
Pairwise balanced designs with $r=\lambda^{2}$
Assuming you meant $r = \lambda^2$ (and also assuming you don't want repeated blocks), a proper approach might be to poke around the properties of $(r,\lambda)$-designs and their constructions to give …
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1
vote
Orthogonal Latin Square 6*6
If I remember correctly, Tarry's original proof used brute force. So there isn't much to mention about the 1900 proof except the fact that it settled Euler's 36 Officer Problem (see Edit2 at the end o …
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14
votes
What are the major open problems in design theory nowaday?
Edit: Since this just became available on arXiv, Peter Keevash solved the existence conjecture of Steiner $t$-designs, which means that what I wrote below a year ago as one of the most important open …
4
votes
Constructing Steiner Triple Systems Algorithmically
Since this thread just got bumped to the front page, historically the very first proof (by T. P. Kirkman, On a Problem in Combinatorics, Cambridge Dublin Math. J. 2 (1847) 191-204, 1847.) of the exist …
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7
votes
Accepted
Constructions of $2-(v,3,3)$-designs
An elementary counting argument shows that $2$-$(v,3,3)$ exists only if $v$ is odd (or, more precisely, for $\lambda \equiv 3 \pmod{6}$ a $2$-$(v,3,\lambda)$ exists only if $v \equiv 1 \pmod{2}$). Thi …
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3
votes
Accepted
Ranks of higher incidence matrices of designs
For the generalization in the first direction, the $p$-rank of the incidence matrix $N$ of an $S(2,k,v)$ is lower bounded by the dimension of the Steinberg module:
$$\operatorname{rank}_2(N)(\operato …
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